Data Interpretation

Growth = percentage change with a direction

When a table shows how one quantity moves year after year, each year-on-year jump is a growth rate: $g = \frac{\text{new} - \text{old}}{\text{old}} \times 100$ Revenue going 80 → 96 crore grows $\frac{16}{80} \times 100 = 20\%$. A negative $g$ is a decline. The base is always the earlier year.

Successive changes multiply — never add

A rise of $a\%$ multiplies the value by $\left(1 + \frac{a}{100}\right)$; a fall of $b\%$ multiplies by $\left(1 - \frac{b}{100}\right)$. Two changes in a row multiply: $\text{net factor} = \left(1 + \frac{a}{100}\right)\left(1 + \frac{b}{100}\right)$

• $+10\%$ then $+10\%$: $1.1 \times 1.1 = 1.21$ → net $+21\%$, not $+20\%$.
• $+25\%$ then $-20\%$: $1.25 \times 0.8 = 1.00$ → net zero — a fall of 20% exactly cancels a rise of 25%.
• Revenue 80 → 96 → 120 → 144: factors $1.2 \times 1.25 \times 1.2 = 1.8$, so the total growth is $+80\%$.

Running backwards

If the final value is known and the initial is asked, divide by the factor — never subtract percentages. $\text{initial} = \frac{\text{final}}{\text{factor}}$ A salary became ₹45,000 after raises of 20% and 25%: original $= \frac{45000}{1.2 \times 1.25} = \frac{45000}{1.5} = ₹30{,}000$. A population of 14,400 after falls of 20% and 10%: original $= \frac{14400}{0.8 \times 0.9} = \frac{14400}{0.72} = 20{,}000$.

Profit percent from an income–expenditure table

$\text{profit}\% = \frac{\text{income} - \text{expenditure}}{\text{expenditure}} \times 100$ The base is expenditure, not income — dividing by income is a standard trap. Highest profit percent is not always the year with highest profit amount: a small year can be very profitable in percentage terms.

Comparing two firms

Compare growth rates in percent, and remember a percent and a percentage point are different. Firm A grows 80% and firm B grows 100% — B is ahead by 20 percentage points, not "by 20%". Also, the bigger absolute increase can belong to the slower grower (a large base).

Worked example: a three-year chain

Revenue goes 80 → 96 → 120 → 144 (₹ crore). Year-on-year: $\frac{16}{80} = 20\%$, $\frac{24}{96} = 25\%$, $\frac{24}{120} = 20\%$. Overall factor $= 1.2 \times 1.25 \times 1.2 = 1.8$, so revenue is up $80\%$ over the span — although the growth rates add to $65\%$. One multiplication settles what addition gets wrong.

Two quick self-checks

• A rise of $a\%$ followed by a fall of $a\%$ always leaves a loss of $\frac{a^2}{100}\%$: 20% up then 20% down nets $-4\%$.
• If the factor chain multiplies to exactly 1, the net change is zero — the setter loves $+25\%$ with $-20\%$ and $+33\tfrac{1}{3}\%$ with $-25\%$.

Quick revision

• $g = \frac{\text{new} - \text{old}}{\text{old}} \times 100$; base = earlier year.
• Chain of changes: multiply factors $(1 \pm \frac{a}{100})$.
• $+25\%$ then $-20\%$ (or $+20\%$ then $-25\%$) = no net change.
• Reverse: divide the final value by the factor.
• Profit% = profit ÷ expenditure × 100.
• Growth comparison in percentage points = subtract the two rates.